Since the introduction of the total variation theory into mathematics in 1881 by Camille Jordan, the theory has not sufficiently found its pride of place among other mathematical theories. This is because even though the total variation criterion has been used regularly, especially in the theory of distributions and the Stieljes integral where a nice solution requires that the functions be of bounded variation, that is, have a finite total variation, the total variation functional itself has not been extensively applied to the solution of mathematical problems. This is in part because of the difficulty encountered in using the functional. In this book, I give a new derivation of the functional and apply it to image edge detection and image quality measurement. Compared to previous methods, the total variation approach gives superior results.